Recently, Nik Weaver proved a quantum analogue of the Ramsey theorem. Weaver's theorem states that for every positive integer k, there exists a positive integer nk such for any quantum channel on the nk × nk matrices, the corresponding quantum graph possesses either a k-dimensional quantum clique or a k-dimensional quantum anti-clique. Quantum anticliques coincide with error-correcting codes, while quantum cliques satisfy a dual property. In this paper we study the quantum graphs of mixed-unitary channels generated by tensor products of Pauli operators, which we call Pauli channels. We show that most quantum graphs arising from Pauli channels have non-trivial quantum cliques or quantum anti-cliques which are stabilizer codes. However, a reformulation of the quantum Ramsey theorem in terms of stabilizer codes and Pauli channels fails. Specifically, for every positive integer n, there exists an n-qubit Pauli channel for which any non-trivial quantum clique or quantum anti-clique fails to be a stabilizer code. We also show that this example is essentially unique, and hence most n-qubit Pauli channels have non-trivial quantum cliques or quantum anti-cliques which are stabilizer codes.